# Solving a Markowitz problem with restrictions (lower and upper bound) to the weights vector

I would like to find a step by step solutionfor the following Markowitx problem. It is a standard markowitz problem. The unique detail (wich is why I am posting this question here) is that there is a upper bound and a lowerbound for the weights vector.

The problem I want to solve:

The detail:

Definitions:

$w$ is the weight vector, $\Sigma$ is the covariance matrix, and $\mu$ is the returns vector.

The constraints $$w \le b_u$$ and $$b_l \le w \Leftrightarrow-w \ge - b_l$$ can all be handled using the Kuhn–Tucker conditions. Numerical solvers exist for these linear constraints too (e.g. this is in R). See als this.
• When you have upper and lower bound constraints there will be additional non-negative Lagrange multipliers $\mu_l\ge 0,\mu_u\ge 0$ (so 2n extra Lagrange mutlipliers) and you will have "complementary slackness" conditions from KKT of the form $\mu_u(b_u-w)=0$. – noob2 Mar 22 '17 at 11:55